Quantum (spin) glasses Leticia F. Cugliandolo LPTHE Jussieu & LPT-ENS Paris France Giulio Biroli (Saclay) Daniel R. Grempel (Saclay) Gustavo Lozano (Buenos Aires) Homero Lozza (Buenos Aires) Constantino da Silva Santos (Porto) Liliana Arrachea (Buenos Aires) Claudio Chamon (Boston Univ.) Thierry Giamarchi (Genève) Malcolm P. Kennett (Cambridge) Pierre Le Doussal (ENS)
Classical glassy arrest Observable (e.g. viscosity) vs control parameter (e.g. temperature) 1e+17 1e+13 Glass 1e+02 (P) η 1e+09 1e+05 Supercoooled liquid Liquid (sec) τ 1e+01 0 200 400 600 800 1000 To Tg Tm T (C) T g at τ = 10 2 s ; T 0 hypothetic static glass transition.
The glassy problem What is the mechanism for the slowing down observed in all types of samples whenever cooling is sufficiently fast? Describe the dynamic crossover at T g from equilibrium dynamics in the (metastable) super-cooled liquid to the non-equilibrium relaxation in the glassy regime. Is there a static phase transition? (Beyond experimental skills...)
Achievements A highly non-trivial mean-field theory (fully-connected disordered spin models self-consistent approximations) mimics T g as a dynamic transition, T o as a static transition T. R. Kirkpatrick, D. Thirumalai & P. Wolynes (80s) describes correctly the dynamics above W. Götze et al (80s) and below the cross-over region at T g LFC & J. Kurchan (93)
Quantum fluctuations Why should one include quantum fluctuations?
SG phase in High Tc SCs, La 1 x Sr 2 Cu 2 O 4 A. Aharony et al, PRL 60, 1330 (88). S. Wakimoto et al, PRB 62, 3547 (00).
Quantum fluctuations What should one change from the mean-field approach to the glassy problem?
Rôle of the environment Classically : trivial statics in the canonical ensemble, dynamics with an appropriate stochastic equation (Langevin, Glauber,...). Quantum mechanically : highly non-trivial. decoherence, localization transition in a dissipative two-level system A. J. Bray & M. A. Moore (84), S. Chakravarty (84). A. J. Leggett et al (80s).
Statics and dynamics of dissipative quantum glassy systems Real-time dynamics : glassy aspects, aging, effective temperatures? Locus and order of the phase transition? Decoherence, localization and interactions. Griffiths phase? Activated or conventional scaling?
Quantum Ising disordered models H S = ij J ij ˆσ z i ˆσ z j + i Γ iˆσ x i + i h iˆσ z i. ˆσ i Pauli matrices. J ij & Γ i quenched random. ij nearest neighbours on the lattice finite d or fully connected mean-field
p-spin interactions H S = i 1 i 2...i p J i1 i 2...i p ˆσ z i 1ˆσ z i 2... ˆσ z i p + i Γ iˆσ x i + i h iˆσ z i, ˆσ i Pauli matrices, J i1 i 2...i p quenched random J 2 i 1 i 2...i p = i 1 i 2... i p all possible p-uplets p! J 2 2N p 1
Spherical model A particle in a random potential Ĥ S = H J + Potential energy i ˆΠ 2 i 2M Kinetic energy [ˆΠ i, Ŝj] = i δ ij Ŝ2 i = N Spherical constraint i Γ 2 /(JM) Canonical commutation rules Strength of quantum fluctuations
Coupling to the environment H E = i ˆσ z i Ñ a=1 c iaˆx a + Ñ a=1 ( ˆp 2 a + m aωa 2 2m a 2 ˆx 2 a ) + counter-term. Ñ indep. quantum harmonic oscillators per spin Distribution of c ia, m a, ω a type of bath. Spectral density I(ω) α ω s ω ω c Dimensionless coupling constant α Cut-off ω c s = 1 Ohmic s < (>)1 sub(super)-ohmic
Reasons for the choice of model Mean-field limit : amenable to analytic calculations. p 3 spin interactions : connection to structural glasses. Spherical case : real-time dynamics is easy to treat. p = 2 spherical model : connection to domain growth as described by the O(N) model with N and d = 3 and the usual model for rotors. Slightly trivial. Oscillator bath : amenable to analytic calculations. 1d model : Simulations of classical 2d system are feasible
Techniques to study the statics Study of the free-energy density Path-integral in imaginary time τ. Gaussian integration over oscillator variables long-range ferromagnetic interactions in the τ direction. Replica trick to average over disorder & replica symmetry breaking (mean-field). Montecarlo simulations of d + 1 classical model (finite d).
Real-time dynamics Two-time dependence 0 t 0 t t =t+t preparation time w m w waiting time measuring time time Correlation C(t + t w, t w ) = [Ô(t + t w), Ô(t w)] + Linear response R(t+t w, t w ) = δ Ô(t+t w) δh(t w ) h=0 = [Ô(t + t w), Ô(t w)]
Real-time dynamics Study of the dynamic generating functional Continuous models Schwinger-Keldysh closed-time path-integral. Gaussian integration over oscillator variables Two-time long-range interactions. Typical initial conditions (ˆρ e ˆρ s and ˆρ s random ) : no need of replica trick to average over disorder. Study of Dyson equations for correlations and responses.
Static & dynamic phase diagram Dissipative Ising p-spin model with p 3 dashed = 1st order, solid = 2nd order thin = static, bold = dynamic α = 0, 0.5 LFC, D. R. Grempel & C. A. da Silva Santos, PRL 85, 2589 (00) LFC, D. R. Grempel, G. Lozano & H. Lozza, PRB 70, 024422 (04)
Real-time dynamics Dissipative spherical p-spin model Symmetric correlation Linear response Comparison between α = 0.2 (PM) and α = 1 (SG) LFC, D. R. Grempel, G. Lozano, H. Lozza & C. da Silva Santos, PRB 66, 014444 (02).
+ -,, +, - * + 0 +. * "! $#&%'$#)( +. * +. / Interactions against lozalization Dissipative spherical p-spin model LFC, D. R. Grempel, G. Lozano, H. Lozza, C. A. da Silva Santos, PRB 66, 014444 (02).
FDT & effective temperatures Classical glassy systems In equilibrium R(t) = 1 T Glasses : χ(t + t w, t w ) dc(t) dt with t 0. breakdown of stationarity & FDT. t+tw t w dt R(t + t w, t ) f (C(t + t w, t w )) 1 T eff C(t + t w, t w ) in the long t w and t t w limit. LFC, J. Kurchan & L. Peliti, PRE 55, 3898 (97)
FDT & effective temperatures Dissipative quantum glassy models The equilibrium FDT R(t + t w, t w ) = becomes idω π e iωt tanh χ(t + t w, t w ) 1 T eff C(t + t w, t w ) ( ) β ω C(ω, t + t w ) 2 t t w if the integral is dominated by ωt 1 and T T eff such that β eff ω 0. LFC & G. Lozano, PRL 80, 4979 (98)
Aging and effective temperatures Dissipative p spin spherical model LFC & G. Lozano, PRL 80, 4979 (98) ; PRB 59, 915 (99)
Dissipative random Ising chain Montecarlo simulations of the 2d classical counterpart 0.4 Static phase diagrams α 0.3 0.2 FM 0.1 PM 0 1 1.5 2 2.5 Tcl T cl Γ LFC, G. S. Lozano & H. Lozza (04).
Dissipative random Ising chain pdfs of local susceptibilities T cl Γ LFC, G. S. Lozano & H. Lozza (04).
Summary Dissipation favors the glassy phase. Real-time dynamics separation of time-scales, aging and effective temperature (d = ). First-order quantum phase transitions (d = ). Griffiths phenomena still present in d = 1 (at least for moderate couplings to the environment). T eff relevant to other quantum non-equilibrium systems. e.g., mesoscopic dissipative conducting ring driven by a time-dependent magnetic field. L. Arrachea & LFC, cond-mat/0407427
Current in driven conducting ring 0.14 0.15 0.12 J dc 0.1 0.08 0.06 0.04 0.1 0.05 0.02 0 0 0.2 0.4 0.6 0.8 1 V L 0.2 0.15 τ B 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Φ J 0.1 0.05 0 0 5 10 15 20 25 30 t L. Arrachea & LFC, cond-mat/0407427
T eff in conducting ring Re[<G K 11 >] Re[<G K 11 >] 0.6 0.4 0.2 0 0.04 0.03 0.02 0.01 0 0.01 0 2 4 6 8 0.02 30 32 34 36 38 τ τ L. Arrachea & LFC, cond-mat/0407427